Make a custom PHP page that catches the data required and stuffs it in the database.
(you can either make a plugin, or a simple standalone page)
You'll need to add some data, either from the session or derrived from the page, containing the username/id, and send this along to the database.
You can simply add a new table to the existing database and add a userid reference column, and the other ones with gamename, score, etc etc
Then, either by a self made plugin, or again a stand alone page, access the database information and show it on the page.
Personally, I'd go with stand alone setups, ... plugins tend to need to get updated, .... often ...